Variations of * Cp*,

p/p

_{∞,}

*and*cf

*over flat-face disk aerospike.*q

_{w}

Open access peer-reviewed chapter

Submitted: October 28th, 2017 Reviewed: January 26th, 2018 Published: June 27th, 2018

DOI: 10.5772/intechopen.74522

The present chapter deals with heat transfer analysis around unspiked and spiked bodies at high speeds. A spike attached to a blunt-nosed body drastically alters its flowfield and influences the aerodynamic heating in a high speed flow. The effect of spike length, shape and spike-nose configuration is numerically studied at zero angle of incidence. The numerical analysis describes overall flowfield features over without and with forward facing spike attached to a blunt body at high speed flow. The shock stand-off distance, sonic line, stagnation point velocity gradient and stagnation point heat flux are analyzed and compared with different aerodisk configurations. It is found that the hemispherical aerodisk experiences high wall heat flux as compared to the flat-faced aerodisk. Numerical and experimental studies reveal that the wall heat flux levels are decreased in the presence of the spikes and aerospike as compared to without attached spiked to the blunt-nose basic configuration.

- aerospike
- blunt body
- CFD
- compressible flow
- convective heat transfer

The shock wave dominates aerodynamic drag and aerodynamic heating at high speeds of a blunt body. The analytical [1] and experimental [2] investigations have shown that for the blunt body noses with a fixed length, a pointed geometry with a blunt nose tip is most beneficial to minimize the wave drag at high speeds. If aerodynamic heating is considered, a large blunt nose radius * R* is preferred since the wall heat flux,

There are several alternative concepts to the aerospike developed such as changing catalyst properties [5], pulse heating in front of the blunt body [6, 7, 8], DC arc discharging [9], electrically heated wire pointing upstream [10], injection of jets of air [11, 12, 13], or plasma technology [14, 15, 16], forward facing cavity [17], self-aligning aerodisk [18], opposite jet injection [19], focused energy deposition [20, 21, 22], non-ablative thermal protection [23] and multiaerodisk attached to the blunt body [24]. Many review articles have appeared [25, 26, 27] to summarize the flowfield characteristics in front of the blunt body to the aerospike.

The features of the high speed flowfield can be delineated through these experimental [28] and numerical [29, 30] studies. Based on these investigations a schematic of the flowfield around the blunt body, the conical, the hemispherical and the flat-face spiked blunt body at zero angle of incidence is delineated in Figure 1. A hemi-spherical portion of the blunt body is accomplished by a bow shock wave as depicted in Figure 1(a). The total pressure loss over the shock leads to a high wave drag. A well-known concept for reducing the impact of the bow shock wave on a blunt body, while keeping a blunt nose, is the aerospike. The simplest aerospike design is a thin rod mounted on the tip of a blunt body as in Figure 1(b). For the aerospike in ideal case, the boundary layer on the rod separates along the whole rod surface due to the pressure rise over the bow shock wave [31]. The separated boundary layer forms a shear layer that is reattached on the blunt nose. Due to the shear layer, the outer supersonic bow is detected and a weaker conical shock is formed instead of the initial bow shock. The conical shock unites with the reattachment shock further downstream. A recirculation zone forms inside the shear surface and shows significantly lower pressure levels compared with the blunt body without an aerospike. In the model of the flat-face and the hemispherical aerodisk, we observe a formation of a bow shock wave ahead of the body as delineated in Figure 1(c) and (d). The flow separation zone is noticed around the root of the spike up to the reattachment point of the flow at the corner of the blunt body. Due to the recirculating region, the pressure at the stagnation region of the blunt body will reduce. However, because of the reattachment of the shear layer on the corner of the blunt body, the pressure near the reattachment point becomes large. However, the reattachment point can depend on the geometrical parameters of the spike or the blunt body configuration. The spike is characterized by a free shear layer, which is formed as a result of the flow separating from spike leading edge and reattaching to the blunt body, essentially bridging the spike. The separating shear layer from the spike leading edge of the aerospike attaches to the blunt body, after entering through an expansion fan at the leading edge corner and a recompression shock at attachment point. The attached shear layer then separates near the trailing edge and generates a separation shock before reattaching point at the trailing edge prior to undergoing flow expansion. The separated boundary layer forms a shear layer that reattaches on the blunt nose.

The above descriptions of the flowfield features show that the flowfield past a spiked blunt body appears to be very complicated and complex and having a number of interesting flow phenomena and characteristics, which have been further studied in order to compute the aerodynamic heating at high speeds.

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## 2. Governing equations

The time-dependent axisymmetric compressible fluid dynamics equations were written in integral form, and the system of equations was augmented by the ideal gas law for numerical simulation [31]. The coefficient of molecular viscosity was calculated using Sutherland’s law. A laminar flow is considered in the numerical simulation which also agrees with Bogdonoff and Vas [32], Fujita and Kubota [33], Yamauchi et al. [34] and Ahmed and Qin [25].

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## 3. Numerical algorithm

### 3.1. Initial and boundary conditions

### 3.2. Computational grid

The flowfield solver uses a finite-volume discretization employing the method of lines. The spatial computational region was divided into a number of finite non-overlapping quadrilateral grids. Thus, the discretized solution to the Navier-Stokes equations results in a set of volume-averaged state variables of mass, momentum and energy, which are in balance with their area-averaged fluxes (inviscid and viscous) across the grid faces [35]. The finite-volume flow solver algorithm written in this way reduces to a central difference scheme and is second-order accurate in space provided that the grid is generated in an orderly manner and is smooth enough. The cell-centered spatial discretization method is non-dissipative [36]; therefore, artificial dissipation terms are added as a blend of a Laplacian and biharmonic operator in an analogous to the second and fourth difference. The artificial dissipation term was added in the algorithm explicitly to prevent numerical oscillations near flow discontinuity to dampen high-frequency undamped modes. Temporal integration was performed using a multistage time-stepping scheme of Jameson et al. [36] and numerical integration used on the Runge–Kutta method. The artificial dissipation is evaluated only at the first stage.

An initial condition corresponding to freestream conditions is considered. All flow quantities were extrapolated at the outer-boundary, and the no-slip condition was imposed on the wall. An isotherm wall condition was used for the wall of the model, that is, a surface temperature of 300 K. The symmetric condition is imposed on the centerline.

One of the controlling factors for the numerical simulation is the proper grid arrangement. The grid points are generated by a homotopy scheme [37]. Mesh-independence tests were performed, taking into consideration the influence of the computational region, the stretching factor to control the mesh intensity in the vicinity of body surface and the number of mesh points in the axial and normal directions. The outer surface of the computational zone is varied about 5–8 times the maximum blunt body diameter * D*. the mesh stretching factor in the radial direction is varied in order to resolve boundary layer. The convergence criterion is based on the difference in density value at any grid point between two successive iterations, that is, ∣

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## 4. Geometrical details of model

### 4.1. Axisymmetric blunt body

### 4.2. Heat shield with forward facing spike

### 4.3. Conical spike attached to blunt body

### 4.4. Conical, disk and flat spiked body

The spherical blunted-cone/flare configuration without spike is illustrated in Figure 2(a). The spherical forebody has R_{N} = 0.51 m, D = 2.03 m, L = 1.67 m, and cone angle of 20°. The flare has a semi-cone angle 15° and is terminated with a right circular cylinder. A close-up view of the computational mesh is shown in Figure 2(b).

The fore and afterbody diameters of the heat shield [38, 39] are 43.26 × 10^{−3} and 35 × 10^{−3} m, respectively, as shown in Figure 3(a). The semi-cone angle of the heat shield is 20°. The hemispherical nose of the forward facing spike has radius of 0.55 × 10^{−3} m and length of the spike is 10 × 10^{−3} m. The other end of spike has diameter of 0.98 × 10^{−3} m and is attached to the blunt spherical cap of the heat shield of radius 8.75 × 10^{−3} m. The computational grid is displayed in Figure 3(b).

The model is axisymmetric, the main body has a hemisphere-cylinder nose and diameter * D* = 7.62 × 10

The dimensions of the spiked blunt body are depicted in Figure 5. The basic body has a hemispherical-cylinder nose and diameter * D* = 4.0 × 10

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## 5. Results and discussions

### 5.1. Flow and heat transfer analysis of axisymmetric blunt body

### 5.2. Heat shield with a forward-facing spike

### 5.3. Conical spike attached to the blunt body

### 5.4. Conical, disk and flat attached to the blunt body

#### 5.4.1. Shock stand-off distance

Δ F D S = 2.8 ρ ∞ ρ 0 E1ρ 2 ρ ∞ = γ + 1 M ∞ 2 γ − 1 M ∞ 2 + 2 E2aρ e ρ ∞ = γ + 1 M ∞ 2 γ − 1 M ∞ 2 + 2 1 + γ − 1 2 γ − 1 M ∞ 2 + 2 2 γ M ∞ 2 − γ + 1 1 / γ − 1 E2bp e p ∞ = γ + 1 M ∞ 2 2 γ / λ − 1 γ + 1 2 γ M ∞ 2 + γ − 1 1 / λ − 1 E2cΔ S D S = 2 ρ ∞ ρ 0 1 + 8 ρ ∞ ρ 0 3 E3Δ R N = 2 b 2 / 3 M ∞ 2 − 1 γ + 1 M ∞ 2 2 / 3 − 1 E4#### 5.4.2. Surface pressure variations

#### 5.4.3. Stagnation point heating and wall heat flux

q w = 0.763 Pr − 0.6 ρ e μ e K ρ w μ w ρ e μ e h e − h w E5du e dx s = 1 R N 2 ρ 2 − ρ ∞ ρ 2 E6q ̇ w , s = 1.83 × 10 − 4 ρ w R n 0.5 V ∞ 3 1 − H w H s E7### 5.5. Heat transfer measurements

q w q S = 2 θ sin θ 1 − 1 γ M ∞ 2 cos 2 θ + 1 1 − 1 γ M ∞ 2 θ 2 − θ sin 4 θ 2 + 1 − cos 4 θ 8 + 4 γ M ∞ 2 θ 2 − θ sin 2 θ + 1 − cos 2 θ 2 E8q t = kρ C p 2 π 2 T t t + ∫ 0 t T t − T τ t − τ 3 / 2 dτ E9

Characteristic features of the flowfield around the hemispherical and the flat-disk aerospike attached to the blunt body at high speeds were investigated with the help of velocity vector plots, density, pressure and Mach contours diagram.

Computed Mach and temperature contours around the unspiked reentry module are shown in Figure 7 for * M* = 2.0–6.0. The Mach contours exhibit the vortices formation at the shoulder region of the module. Behaviors of the flowfield around the blunt body at supersonic speeds shows the formation of the bow shock wave ahead of the blunt body; the wake, and the recompression shock waves coming out from the neck point are observed in the Mach contours. The flow expands at the base region and is followed by the recompression shock downstream of the base. Then, the flow progress in the wake region. The flow ground of the capsule is divided into regions inside and outside of the flow recirculation zone, and two flow zones are separated by the shear layer. From the temperature contours, a rapid raise of temperature with the increase of

Figure 8 shows the pressure coefficient [* Cp* =

Figure 9 depicts the close-up view of the velocity vector plots and Mach contours unspiked and spiked over the heat shield [39]. We can visualize from the vector plot in Figure 9(a) the interaction between bow shock wave and reattachment shock. The recirculation region behaves as if it has a spike boundary. A significant flowfield is found around the unspiked and spiked heat shield as depicted in Figure 9(b). Figure 10 depicts the surface pressure coefficient and wall heat flux variation along blunt body surface for M_{∞} = 2.0. A very high-pressure peak can be seen in the * Cp* variation. The wall heat flux variation along the model exhibits similar characteristics as the

Interaction between the conical oblique shock wave starting from the tip of the spike and the reattachment shock wave of blunt body is observed in the pressure, density and Mach contours in Figure 11. The reflected reattachment wave and shear layer from the interaction are shown behind the reattachment shock wave. A large separated region is found in front of the blunt body and the shear layer, the boundary of the separated region is visible. The variation of * p*/

The computed density contour plots over the conical, the hemispherical and the flat-faced aerospiked configurations are shown in Figure 13(a), (b) and (c), respectively. The separated shear layer and the recompression shock from the reattachment point on the corner of the unspiked blunt body are observed in the flowfield region of contour diagram. The bow shock wave in front of the aerospike disk will decrease the aerodynamic drag as compared to the case the unspiked body as observed in the contour plots. In the fore-region of the aerodisk, the flow velocity decreases after the bow shock wave. At the corner of the aerodisk, the flow turns and expands rapidly, the boundary layer separates, gives a free-shear layer that separates the recirculating flow zone after the base region to the outer region. For the case of a button type spike flying at hypersonic speeds, a detached bow shock wave is observed in front of the spike which is appeared normal at to the body axis [29]. Since the flow behind the normal shock is subsonic, simple continuity considerations may show that the shock-detachment distance and stagnation-velocity gradient are essentially a function of * ρ*/

The gas is considered thermally and calorically perfect. The ratio of the flow properties across the normal shock wave [41, 42] can be calculated as a function of _{∞} and * γ*, the relations are

The effects of the subsonic flow on the hemispherical and the flat-face disk bodies have been investigated by Truitt [43]. The nomenclature is illustrated in Figure 14(a). The freestream flow passes through the normal portion of the shock wave reaching state * 2* then decelerates isentropically to state

The values of * ∆* and

where the value of * b* is taken as 0.14 [45]. The bow shock stand-off distance Δ/R

The pressure coefficient distribution on the spiked blunt-body with other aerospikes configurations is given in Figure 15. The * x/R = 0* is the location of the spike tip, where

Tables 1 and 2 depict the variation of pressure coefficient, non-dimensional pressure, skin friction coefficient and wall heat flux over the spike surface facing the flow direction along the spike. The * s/D* is measured from the stagnation point. The

/_{S} | _{∞} | , W/m_{w}^{2} | ||
---|---|---|---|---|

.00000E + 00 | .14776E + 01 | .38234E + 02 | .42341E-05 | .21158E + 06 |

.21500E-02 | .15140E + 01 | .39153E + 02 | .11539E-03 | .55592E + 07 |

.51000E-02 | .15278E + 01 | .39502E + 02 | .68686E-04 | .30266E + 07 |

.88500E-02 | .15021E + 01 | .38854E + 02 | .55081E-04 | .21678E + 07 |

.13500E-01 | .14458E + 01 | .37433E + 02 | .49153E-04 | .17160E + 07 |

.18950E-01 | .13656E + 01 | .35414E + 02 | .45983E-04 | .14250E + 07 |

.25200E-01 | .12628E + 01 | .32823E + 02 | .43713E-04 | .12121E + 07 |

.32300E-01 | .11468E + 01 | .29898E + 02 | .40917E-04 | .10449E + 07 |

.40250E-01 | .10324E + 01 | .27016E + 02 | .36909E-04 | .90977E + 06 |

.49000E-01 | .92245E + 00 | .24246E + 02 | .33641E-04 | .79750E + 06 |

.58600E-01 | .82295E + 00 | .21738E + 02 | .32589E-04 | .71001E + 06 |

.69000E-01 | .71613E + 00 | .19046E + 02 | .33443E-04 | .64654E + 06 |

.80250E-01 | .59834E + 00 | .16078E + 02 | .35029E-04 | .59998E + 06 |

.92300E-01 | .48029E + 00 | .13103E + 02 | .35604E-04 | .55673E + 06 |

/_{S.} | _{∞} | , W/m_{w}^{2} | ||
---|---|---|---|---|

0 | 1.8987 | 48.846 | 1.3293e-06 | 326,130 |

0.005 | 1.8974 | 48.814 | 5.7879e-05 | 1.4549e + 07 |

0.01 | 1.8906 | 48.643 | 2.7381e-05 | 7.2722e + 06 |

0.015 | 1.8798 | 48.372 | 1.6742e-05 | 4.8444e + 06 |

0.02 | 1.8669 | 48.046 | 1.1241e-05 | 3.6302e + 06 |

0.025 | 1.8529 | 47.694 | 7.9517e-06 | 2.9018e + 06 |

0.03 | 1.8386 | 47.333 | 5.8737e-06 | 2.4167e + 06 |

0.035 | 1.8244 | 46.974 | 4.5396e-06 | 2.0707e + 06 |

0.04 | 1.8102 | 46.616 | 3.6880e-06 | 1.8115e + 06 |

0.045 | 1.7957 | 46.253 | 3.1615e-06 | 1.6101e + 06 |

0.05 | 1.7807 | 45.873 | 2.8655e-06 | 1.4489e + 06 |

0.055 | 1.7642 | 45.458 | 2.7391e-06 | 1.3167e + 06 |

0.06 | 1.7449 | 44.972 | 2.7632e-06 | 1.2057e + 06 |

0.065 | 1.7221 | 44.396 | 3.0004e-06 | 1.1111e + 06 |

0.07 | 1.6957 | 43.731 | 3.5721e-06 | 1.0305e + 06 |

0.075 | 1.6573 | 42.765 | 4.8788e-06 | 960,610 |

0.08 | 1.5942 | 41.173 | 7.8607e-06 | 897,760 |

0.085 | 1.5726 | 40.629 | 1.0038e-05 | 853,270 |

0.09 | 1.3785 | 35.737 | 1.1276e-05 | 826,300 |

0.095 | 1.2663 | 32.910 | 1.0595e-05 | 855,090 |

0.1 | 0.5229 | 14.177 | 1.2575e-05 | 708,300 |

The nose tip of a high-speed vehicle usually is hemi-spherically in shape. Consequently, a normal detached shock is formed in front of the stagnation point as depicted in Figure 14(a) which extends around the body as a curved oblique shock. The shock wave stands in front of the blunt body and forms a region of subsonic flow around the stagnation region and sonic line. The flow in the shock is at low subsonic speeds in the stagnation region and accelerates to sonic speeds in the shoulder region. For flat-nosed body, the detached bow shock wave of the nose is particularly normal to the body axis. A compressible subsonic region is formed between the body and the shock which is a function of density ratio across the normal shock. The shock-detachment distance Δ and stagnation-velocity gradient * K* are essentially functions of density ratio across the normal shock [41, 42]. One of the areas of concern is the stagnation point heating of a spiked and unspiked body, when the incoming high-velocity flow is come to stagnation on the wall by a normal shock and adiabatically compression process. The problem now becomes finding out the heat flux in the vicinity of the stagnation point. It requires a solution of the entire flowfield from shock to the body.

The inviscid flowfield in the vicinity of the stagnation point is described in a fluid dynamics sense as the conversion of a unidirectional high-velocity stream by a normal shock wave into a high temperature subsonic layer, which is taken to be inviscid and incompressible [43]. The heat transfer rate is directly proportional to the enthalpy gradient on the body surface and square root of the velocity gradient, (* β* =

The stagnation point velocity gradient can be written as non-dimension parameter as (* K* =

It is important to mention here that the sphere shows the greatest change in velocity gradient as compared to the flat disk. The magnitude of the stagnation-velocity gradient indicates the maximum heat transfer rate. The disk or flat plate with free streamlines will experience the lowest stagnation wall temperature of blunt body. The value * K* is considered as 0.3 [50]. The hemispherical aerospike exhibits the significant changes in the magnitude of

where * R* is nose radius in m,

Experiments [51] were carried out in shock tunnel to measure the heat flux on the blunt body attached with the hemispherical aerospike of * L*/

The wall heat flux values are non-dimensionlised with the stagnation heat flux value. The temperature variations on the model surface that can be recorded using platinum thin film gauge can be used to find the heat transfer rate to the model by using the following expression [53]:

Evaluate above equation by applying approximate numerical schemes. Numerical methods involve division of the time interval (0, * t*) into a finite number of increments and the evaluation of the integrand at each of the division points. The heat flux over the basic configuration with and without spiked for a fixed

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## 6. Conclusions

The flowfield around a forward facing a hemispherical and a flat disk spike attached to blunt-nosed body has been numerically simulated at high speeds at zero angle of attack. The flow visualizations were done using the velocity vector and contour plots in order to analyze the influence of the shape of the spike on the drag reduction and wall heat flux. The formation of the bow shock wave is observed over the unspiked and spiked blunt body. Different flow separation zones depend on the shape of the spike attached to the blunt body. The stand-off distances of the bow shock wave for the hemispherical and the flat aerospike are compared with the analytical solutions and are seen in good agreement. The variations of surface pressure, the skin friction coefficient and the wall heat flux along the surface of the spike facing the flow direction is significantly influenced by the geometrical shape of the spike. The density and pressure ratio and the heat flux at the stagnation point are computed and compared with the analytical results. The numerical analysis delineates complete flowfield information over the unspiked and the spiked blunt-body surface including the bow shock, shock stand-off distance shock, sonic line and stagnation point velocity gradient.

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## Nomenclature

Cf | skin friction coefficient |

Cp | pressure coefficient |

D | cylinder diameter |

L | length of the spike |

M | Mach number |

q | heat flux |

p | pressure |

RN | radius of the spherical body |

s | distance along the surface of the spike |

T | temperature |

t | time |

x, r | coordinate direction |

γ | ratio of specific heats |

ρ | density |

τ | dummy line variable |

Δ | shock stand-off distance |

Subscripts. | |

e | edge of the boundary layer |

o | stagnation |

w | wall |

∞ | freestream condition |

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Submitted: October 28th, 2017 Reviewed: January 26th, 2018 Published: June 27th, 2018

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